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Monday, 25 March 2013

The Addition Formulas

The fundamental identities are very important for the analysis of trigonometric expressions and functions but they are a direct result of the intimate relation between trigonometry and geometry. The power behind the algebraic nature of trigonometry is hidden and can be measured only with the addition formulas

and
Of course, we used the fact that

Example. verify the identityAnswer. We have
which gives
But
and since
and , we get finally

Remark. In general it is good to check whether the given formula is correct. One way to do that is to substitute some numbers for the variables. For example, if we take a=b = 0, we get
or we may take . In this case we have

Example. Find the exact value ofAnswer. We have
Hence, using the additions formulas for the cosine function we get
Since
we get

Example. Find the exact value for

Answer. We have
Since
we get
Finally we have

Remark. Using the addition formulas, we generate the following identities